2
Polynomials
Degree · Zeroes · Graphs · Relation between zeroes and coefficients
Algebra unit
20 marks
20 marks
Definitions
- Polynomial in x:
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ⋯ + a₁x + a₀, where powers of x are whole numbers (0, 1, 2, …).
x² + √x and x + 1x are not polynomials. - Degree: the highest power of x in p(x).
- Zero of a polynomial: a number k is a zero of p(x) if p(k) = 0.
- Zeroes vs roots: zeroes are for a polynomial p(x); roots are for the equation p(x) = 0. Same values.
| Degree | Name | General form | Max zeroes |
|---|---|---|---|
| 0 | Constant | k | 0 |
| 1 | Linear | ax + b | 1 |
| 2 | Quadratic | ax² + bx + c | 2 |
| 3 | Cubic | ax³ + bx² + cx + d | 3 |
In every form, a ≠ 0. By number of terms: monomial (1), binomial (2), trinomial (3).
Must remember
A polynomial of degree n has at most n zeroes.
Zero of linear ax + b is x = −ba.
Geometrical meaning of zeroes
Key idea
The zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) cuts or touches the x-axis. Number of zeroes = number of such points (yellow dots below).
The graph of a quadratic polynomial is a curve called a parabola. a > 0 ⇒ opens upward (∪), a < 0 ⇒ opens downward (∩).
Relation between zeroes and coefficients
Quadratic ax² + bx + c · zeroes α, β
Sum of zeroesα + β = −ba
Product of zeroesαβ = ca
Cubic ax³ + bx² + cx + d
Sumα + β + γ = −ba
Pairwise productsαβ + βγ + γα = ca
Productαβγ = −da
Quadratic from zeroes
p(x) = k [x² − (α + β)x + αβ]
i.e. k [x² − (Sum) x + (Product)], k = any non-zero real constant
Exam tip — quick checks
- Zeroes are reciprocals (αβ = 1) ⇒ c = a
- Zeroes are equal & opposite (α + β = 0) ⇒ b = 0
- One zero is 0 ⇒ c = 0
- Always verify the relation after finding zeroes — it is often asked.
Identities in α and β (very useful)
11α + 1β = α + βαβ
2α² + β² = (α + β)² − 2αβ
3αβ + βα = (α + β)² − 2αβαβ
4(α − β)² = (α + β)² − 4αβ
5α − β = √(α + β)² − 4αβ
6α³ + β³ = (α + β)³ − 3αβ(α + β)
7α²β + αβ² = αβ(α + β)
81α² + 1β² = (α + β)² − 2αβ(αβ)²
Class 10 Maths · Complete Formula Sheet · CBSEBy Mandeep · Green Board
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